Define the linear transformation

To find the range of T , apply T to the elements of a spanning set for C3 as suggested in Theorem SSRLT. We will use the standard basis vectors (Theorem SUVB).

Each of these vectors is a scalar multiple of the others, so we can toss two of them in reducing the spanning set to a linearly independent set (or be more careful and apply Theorem BCS on a matrix with these three vectors as columns). The result is the basis of the range,

Since (T ) has dimension 1, and the codomain has dimension 2, they cannot be equal. So Theorem RSLT says T is not surjective.